Gil has struggled to establish himself in North London after joining from Sevilla at the start of the 2021/22 season with zero Premier League starts to his name.
The 21-year-old was tipped for a bright future in the Premier League, but Spurs boss Antonio Conte still appears to be unconvinced by his ability to make an impact in English football.
Conte ratified a loan switch to Valencia at the start of 2022 but he managed just nine league starts in the second half of the campaign.
According to reports from Diario AS, Conte is open to allowing him to leave the club again this summer, with La Real monitoring the situation with interest.
The Basque side are open to either a loan or a permanent deal, however, Spurs are likely to insist on a purchase clause, if he joins La Real on loan in 2022/23.